category theory

# Contents

## Definition

###### Definition

$\left(F⊣G⊣H\right):C\to D$( F\dashv G\dashv H) : C \to D

is a triple of functors $F,H:C\to D$ and $G:D\to C$ together with adjunction data $F⊣G$ and $G⊣H$.

## Properties

###### Note

The two adjunctions imply of course that $G$ preserves all limits and colimits that exist in $D$.

###### Note

$\left(F⊣G⊣H\right):C\to D$(F \dashv G \dashv H) : C \to D

gives rise to an adjoint pair

$\left(GF⊣GH\right):C\to C$(G F \dashv G H) : C \to C

consisting of a monad $GF$ left adjoint to the comonad $GH$ on $C$;

as well as an adjoint pair

$\left(FG⊣HG\right):D\to D\phantom{\rule{thinmathspace}{0ex}}.$( F G \dashv H G ) : D \to D \,.

###### Proposition

For an adjoint triple $F⊣G⊣H$ we have that $F$ is fully faithful precisely if $H$ is fully faithful.

###### Proof

By a basic property of adjoint functors we have that

• the left adjoint $F$ being full and faithful is equivalent to the unit $\mathrm{Id}\to GF$ being a natural isomorphism;

• the right adjoint $H$ being full and faithful is equivalent to the counit $GH\to \mathrm{Id}$ being a natural isomorphism.

Moreover, by note 2 and the fact that adjoints are unique up to isomorphism, we have that $GF$ is isomorphic to the identity precisely if $GH$ is.

Finally, by a standard fact about adjoint functors (for instance (Elephant, lemma 1.1.1) $GH$ is isomorphic to the identity precisely if it is so by the adjunction unit.

The preceeding proposition is folklore; perhaps its earliest appearance in print is (DT, Lemma 1.3). A slightly shorter proof is in (KL, Prop. 2.3). Both proofs explicitly exhibit an inverse to the counit $GH\to \mathrm{Id}$ or the unit $\mathrm{Id}\to GF$ given an inverse to the other (which could be extracted by beta-reducing the above, slightly more abstract argument).

In the situation of Proposition 1, we say that $F⊣G⊣H$ is a fully faithful adjoint triple. This is often the case when $D$ is a category of “spaces” structured over $C$, where $F$ and $H$ construct “discrete” and “codiscrete” spaces respectively.

For instance, if $G:D\to C$ is a topological concrete category, then it has both a left and right adjoint which are fully faithful. Not every fully faithful adjoint triple is a topological concrete category (among other things, $G$ need not be faithful), but they do exhibit certain similar phenomena. In particular, we have the following.

###### Proposition

Suppose $\left(F⊣G⊣H\right):C\to D$ is an adjoint triple in which $F$ and $H$ are fully faithful, and suppose that $C$ is cocomplete. Then $G$ admits final lifts for small $G$-structured sinks.

###### Proof

Let $\left\{G\left({S}_{i}\right)\to X\right\}$ be a small sink in $C$, and consider the diagram in $D$ consisting of all the ${S}_{i}$, all the counits $\epsilon :FG\left({S}_{i}\right)\to {S}_{i}$ (where $F$ is the left adjoint of $G$), and all the images $FG\left({S}_{i}\right)\to F\left(X\right)$ of the morphisms making up the sink. The colimit of this diagram is preserved by $G$ (since it has a right adjoint as well). But the image of the diagram consists essentially of just the sink itself (since $F$ is fully faithful, $G\left(\epsilon \right)$ is an isomorphism), and its colimit is $X$; hence the colimit of the original diagram is a lifting of $X$ to $D$ (up to isomorphism). It is easy to verify that this lifting has the correct universal property.

Thus, we can talk about objects of $D$ having the weak structure or strong structure induced by any small collection of maps.

###### Corollary

In the situation of Proposition 2, $G$ is a (Street) opfibration. If it is also an isofibration, then it is a Grothendieck opfibration.

###### Proof

A final lift of a singleton sink is precisely an opcartesian arrow.

Dually, of course, if $C$ is complete, then $G$ admits initial lifts for small $G$-structured cosinks and is a fibration.

In particular, the proposition and its corollary apply to a cohesive topos, and (suitably categorified) to a cohesive (∞,1)-topos.

## Examples

### Special cases

• An adjoint triple $F⊣G⊣H$ is Frobenius if $F$ is naturally isomorphic to $H$. See Frobenius functor.

• An affine morphism is an adjoint triple of functors in which the middle term is conservative. For example, any affine morphism of schemes induce an affine triples of functors among the categories of quasicoherent modules.

• An adjoint triple of functors among ${A}_{\infty }$- or triangulated functor?s with certain additional structure is called spherical . See e.g. (Anno). The main examples come from Serre functors in a Calabi-Yau category context.

### Specific examples

• Given any ring homomorphism ${f}^{\circ }:R\to S$ (in commutative case dual to an affine morphism $f:\mathrm{Spec}S\to \mathrm{Spec}R$ of affine schemes), there is an adjoint triple ${f}^{*}⊣{f}_{*}⊣{f}^{*}$ where ${f}^{*}:{}_{R}\mathrm{Mod}\to {}_{S}\mathrm{Mod}$ is an extension of scalars, ${f}_{*}:{}_{S}\mathrm{Mod}\to {}_{R}\mathrm{Mod}$ the restriction of scalars and ${f}^{!}:M↦{\mathrm{Hom}}_{R}\left({}_{R}S,{}_{R}M\right)$ its right adjoint. This triple is affine in the above sense.

## References

Some remarks on adjoint triples are in

On spherical triples see

Generalities are in

Proofs of the folklore Proposition 1 can be found in

• Roy Dyckhoff and Walter Tholen, “Exponentiable morphisms, partial products, and pullback complements”, JPAA 49 (1987), 103–116.
• G.M. Kelly and F.W. Lawvere, “On the complete lattice of essential localizations”, Bulletin de la Société Mathématique de Belgique, Série A, v. 41 no 2 (1989) 289–319.